.. _sphx_glr_api_gallery_fluids_continuum_mechanics:

Continuum Mechanics
-------------------

Linear continuum mechanics beyond flow. Fourier's heat equation smooths a
temperature field mode by mode. The explicit, Crank-Nicolson, and spectral
solvers show how stability and accuracy trade off. Sound in a pipe or a
room rings at a discrete set of standing-wave frequencies, recovered here
from a single pressure pulse. A pressurized thick-walled cylinder, solved
with plane-strain finite elements, reproduces Lamé's closed-form stresses.


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    <div class="sphx-glr-thumbcontainer" tooltip="Small pressure disturbances in air travel at c = \sqrt{\gamma p/\rho} and obey the wave equation. In a pipe, reflections at the two ends allow only a discrete set of frequencies to persist: its standing modes. An end closed by a rigid cap reflects a pressure maximum. An open end, to a good approximation, fixes the pressure at ambient. As a result a pipe open at both ends (a flute) sounds all harmonics f_n = nc/2L, while a pipe closed at one end (a clarinet, roughly) sounds only the odd ones, f_n = (2n-1)c/4L, an octave lower.">

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  .. image:: /api/gallery/fluids/continuum_mechanics/images/thumb/sphx_glr_plot_acoustic_pipe_and_room_modes_thumb.png
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  :doc:`/api/gallery/fluids/continuum_mechanics/plot_acoustic_pipe_and_room_modes`

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      <div class="sphx-glr-thumbnail-title">Sound in pipes and rooms: standing modes from a single pulse</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Joseph Fourier&#x27;s Théorie analytique de la chaleur (1822) wrote heat conduction as">

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  :doc:`/api/gallery/fluids/continuum_mechanics/plot_fourier_heat_equation`

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      <div class="sphx-glr-thumbnail-title">Fourier's heat equation: every mode decays on its own</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="A gun barrel, a pressure vessel or a hydraulic pipe is a thick cylinder with pressure p inside. Gabriel Lamé (1852) found its stresses in closed form:">

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  .. image:: /api/gallery/fluids/continuum_mechanics/images/thumb/sphx_glr_plot_lame_pressurized_cylinder_thumb.png
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  :doc:`/api/gallery/fluids/continuum_mechanics/plot_lame_pressurized_cylinder`

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      <div class="sphx-glr-thumbnail-title">Lamé's thick-walled cylinder: plane-strain finite elements</div>
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.. toctree::
   :hidden:

   /api/gallery/fluids/continuum_mechanics/plot_acoustic_pipe_and_room_modes
   /api/gallery/fluids/continuum_mechanics/plot_fourier_heat_equation
   /api/gallery/fluids/continuum_mechanics/plot_lame_pressurized_cylinder